# Tensorization of the strong data processing inequality for quantum chi-square divergences

Yu Cao1 and Jianfeng Lu1,2

1Department of Mathematics, Duke University, Box 90320, Durham NC 27708, USA
2Department of Physics and Department of Chemistry, Duke University, Box 90320, Durham NC 27708, USA

### Abstract

It is well-known that any quantum channel $\mathcal{E}$ satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum $\chi^2_{\kappa}$ divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states $\rho$ and $\sigma$ does not increase under the action of any quantum channel $\mathcal{E}$. For a fixed channel $\mathcal{E}$ and a state $\sigma$, the divergence between output states $\mathcal{E}(\rho)$ and $\mathcal{E}(\sigma)$ might be strictly smaller than the divergence between input states $\rho$ and $\sigma$, which is characterized by the strong data processing inequality (SDPI). Among various input states $\rho$, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum $\chi^2_{\kappa_{1/2}}$ divergence for arbitrary quantum channels and also for a family of $\chi^2_{\kappa}$ divergences (with $\kappa \ge \kappa_{1/2}$) for arbitrary quantum-classical channels.

### ► References

[1] Venkat Anantharam, Amin Gohari, Sudeep Kamath, and Chandra Nair. On maximal correlation, hypercontractivity, and the data processing inequality studied by Erkip and Cover, Apr 2013. arXiv:1304.6133.
arXiv:1304.6133

[2] Salman Beigi. A new quantum data processing inequality. J. Math. Phys., 54 (8): 082202, 2013. 10.1063/​1.4818985.
https:/​/​doi.org/​10.1063/​1.4818985

[3] Salman Beigi, Nilanjana Datta, and Cambyse Rouzé. Quantum reverse hypercontractivity: its tensorization and application to strong converses, Apr 2018. arXiv:1804.10100.
arXiv:1804.10100

[4] Hong-Yi Chen, György Pál Gehér, Chih-Neng Liu, Lajos Molnár, Dániel Virosztek, and Ngai-Ching Wong. Maps on positive definite operators preserving the quantum $\chi^2_{\alpha}$-divergence. Lett. Math. Phys., 107 (12): 2267–2290, 2017. 10.1007/​s11005-017-0989-0.
https:/​/​doi.org/​10.1007/​s11005-017-0989-0

[5] Man-Duen Choi, Mary Beth Ruskai, and Eugene Seneta. Equivalence of certain entropy contraction coefficients. Linear Algebra Appl., 208-209: 29–36, 1994. 10.1016/​0024-3795(94)90428-6.
https:/​/​doi.org/​10.1016/​0024-3795(94)90428-6

[6] Joel E. Cohen, Yoh Iwasa, Gh. Rautu, Mary Beth Ruskai, Eugene Seneta, and Gh. Zbaganu. Relative entropy under mappings by stochastic matrices. Linear Algebra Appl., 179: 211–235, 1993. 10.1016/​0024-3795(93)90331-H.
https:/​/​doi.org/​10.1016/​0024-3795(93)90331-H

[7] Toby Cubitt, Michael Kastoryano, Ashley Montanaro, and Kristan Temme. Quantum reverse hypercontractivity. J. Math. Phys., 56 (10): 102204, 2015. 10.1063/​1.4933219.
https:/​/​doi.org/​10.1063/​1.4933219

[8] Fumio Hiai and Mary Beth Ruskai. Contraction coefficients for noisy quantum channels. J. Math. Phys., 57 (1): 015211, 2016. 10.1063/​1.4936215.
https:/​/​doi.org/​10.1063/​1.4936215

[9] Fumio Hiai, Hideki Kosaki, Dénes Petz, and Mary Beth Ruskai. Families of completely positive maps associated with monotone metrics. Linear Algebra Appl., 439 (7): 1749–1791, October 2013. 10.1016/​j.laa.2013.05.012.
https:/​/​doi.org/​10.1016/​j.laa.2013.05.012

[10] Michael J. Kastoryano and Kristan Temme. Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys., 54 (5): 052202, 2013. 10.1063/​1.4804995.
https:/​/​doi.org/​10.1063/​1.4804995

[11] Christopher King. Hypercontractivity for semigroups of unital qubit channels. Commun. Math. Phys., 328 (1): 285–301, May 2014. 10.1007/​s00220-014-1982-4.
https:/​/​doi.org/​10.1007/​s00220-014-1982-4

[12] Andrew Lesniewski and Mary Beth Ruskai. Monotone Riemannian metrics and relative entropy on noncommutative probability spaces. J. Math. Phys., 40 (11): 5702–5724, 1999. 10.1063/​1.533053.
https:/​/​doi.org/​10.1063/​1.533053

[13] Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel. On quantum Rényi entropies: A new generalization and some properties. J. Math. Phys., 54 (12): 122203, December 2013. 10.1063/​1.4838856.
https:/​/​doi.org/​10.1063/​1.4838856

[14] Dénes Petz. Sufficiency of channels over von Neumann algebras. Q. J. Math., 39 (1): 97–108, 1988. 10.1093/​qmath/​39.1.97.
https:/​/​doi.org/​10.1093/​qmath/​39.1.97

[15] Dénes Petz and Mary Beth Ruskai. Contraction of generalized relative entropy under stochastic mappings on matrices. Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 01 (1): 83–89, 1998. 10.1142/​S0219025798000077.
https:/​/​doi.org/​10.1142/​S0219025798000077

[16] Yury Polyanskiy and Yihong Wu. Dissipation of information in channels with input constraints. IEEE T. Inform. Theory, 62 (1): 35–55, 2016. 10.1109/​TIT.2015.2482978.
https:/​/​doi.org/​10.1109/​TIT.2015.2482978

[17] Yury Polyanskiy and Yihong Wu. Strong data-processing inequalities for channels and Bayesian networks. In Eric Carlen, Mokshay Madiman, and Elisabeth M. Werner, editors, Convexity and Concentration, The IMA Volumes in Mathematics and its Applications, pages 211–249. Springer New York, 2017. 10.1007/​978-1-4939-7005-6_7.
https:/​/​doi.org/​10.1007/​978-1-4939-7005-6_7

[18] Maxim Raginsky. Strong data processing inequalities and $\Phi$-Sobolev inequalities for discrete channels. IEEE T. Inform. Theory, 62 (6): 3355–3389, June 2016. 10.1109/​TIT.2016.2549542.
https:/​/​doi.org/​10.1109/​TIT.2016.2549542

[19] Cambyse Rouzé and Nilanjana Datta. Concentration of quantum states from quantum functional and transportation cost inequalities. J. Math. Phys., 60 (1): 012202, 2019. 10.1063/​1.5023210.
https:/​/​doi.org/​10.1063/​1.5023210

[20] Mary Beth Ruskai. Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy. Rev. Math. Phys., 06 (5): 1147–1161, 1994. 10.1142/​S0129055X94000407.
https:/​/​doi.org/​10.1142/​S0129055X94000407

[21] K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, and F. Verstraete. The $\chi^2$-divergence and mixing times of quantum Markov processes. J. Math. Phys., 51 (12): 122201, 2010. 10.1063/​1.3511335.
https:/​/​doi.org/​10.1063/​1.3511335

[22] Ramon van Handel. Probability in high dimension, 2016. available at https:/​/​web.math.princeton.edu/​ rvan/​APC550.pdf.
https:/​/​web.math.princeton.edu/​~rvan/​APC550.pdf

[23] Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2013. 10.1017/​CBO9781139525343.
https:/​/​doi.org/​10.1017/​CBO9781139525343

[24] Aolin Xu and Maxim Raginsky. Converses for distributed estimation via strong data processing inequalities. In 2015 ISIT, pages 2376–2380, 2015. 10.1109/​ISIT.2015.7282881.
https:/​/​doi.org/​10.1109/​ISIT.2015.7282881

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